# Properties

 Label 1710.2.p.d Level $1710$ Weight $2$ Character orbit 1710.p Analytic conductor $13.654$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1710.p (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.6544187456$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 153 x^{16} + 6416 x^{12} + 78648 x^{8} + 19120 x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 570) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{4} q^{4} -\beta_{6} q^{5} + ( \beta_{6} - \beta_{9} ) q^{7} -\beta_{11} q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + \beta_{4} q^{4} -\beta_{6} q^{5} + ( \beta_{6} - \beta_{9} ) q^{7} -\beta_{11} q^{8} + \beta_{13} q^{10} + \beta_{1} q^{11} + ( -\beta_{12} + \beta_{13} + \beta_{19} ) q^{13} + ( -\beta_{13} - \beta_{15} ) q^{14} - q^{16} + ( -\beta_{3} + \beta_{5} + \beta_{7} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{17} + ( -\beta_{4} + \beta_{17} + \beta_{19} ) q^{19} + \beta_{7} q^{20} + \beta_{3} q^{22} + ( -2 - 2 \beta_{4} + \beta_{6} + \beta_{9} ) q^{23} + ( \beta_{1} + \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{16} + \beta_{17} ) q^{25} + ( \beta_{1} + \beta_{7} + \beta_{9} ) q^{26} + ( \beta_{5} - \beta_{7} ) q^{28} + ( \beta_{3} - \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{18} + \beta_{19} ) q^{29} + ( -\beta_{3} - \beta_{8} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{18} + \beta_{19} ) q^{31} -\beta_{2} q^{32} + ( -\beta_{10} - \beta_{12} + \beta_{14} - \beta_{18} ) q^{34} + ( -\beta_{1} + 4 \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{16} - \beta_{17} ) q^{35} + ( -\beta_{3} - \beta_{8} + \beta_{10} + 2 \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{37} + ( \beta_{11} - \beta_{18} ) q^{38} + \beta_{14} q^{40} + ( \beta_{2} + \beta_{3} - \beta_{11} - \beta_{12} - \beta_{14} - \beta_{19} ) q^{41} + ( 3 + 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{16} - \beta_{17} ) q^{43} + ( -\beta_{1} + \beta_{8} + \beta_{10} ) q^{44} + ( -2 \beta_{2} + 2 \beta_{11} - \beta_{13} + \beta_{15} ) q^{46} + ( -2 \beta_{1} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{47} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{17} + \beta_{19} ) q^{49} + ( -\beta_{8} - \beta_{11} + \beta_{13} - \beta_{15} - \beta_{18} + \beta_{19} ) q^{50} + ( \beta_{3} + \beta_{14} + \beta_{15} ) q^{52} + ( -\beta_{8} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{18} ) q^{53} + ( \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{16} - \beta_{17} - \beta_{19} ) q^{55} + ( -\beta_{12} - \beta_{14} ) q^{56} + ( -\beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{16} - \beta_{17} ) q^{58} + ( \beta_{2} + \beta_{3} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} - \beta_{18} + \beta_{19} ) q^{59} + ( 2 + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} ) q^{61} + ( 2 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{62} -\beta_{4} q^{64} + ( -3 \beta_{2} + \beta_{3} - \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{15} + 2 \beta_{19} ) q^{65} + ( -2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{67} + ( \beta_{6} + \beta_{9} + \beta_{16} - \beta_{17} ) q^{68} + ( \beta_{8} - 4 \beta_{11} - \beta_{13} + \beta_{15} + \beta_{18} - \beta_{19} ) q^{70} + ( 2 \beta_{13} - 2 \beta_{15} ) q^{71} + ( -1 - \beta_{4} - 2 \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{73} + ( \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{17} + \beta_{19} ) q^{74} + ( 1 + \beta_{16} ) q^{76} + ( 2 - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{16} + 2 \beta_{17} + 2 \beta_{19} ) q^{77} + ( \beta_{3} - \beta_{10} - \beta_{18} + \beta_{19} ) q^{79} + \beta_{6} q^{80} + ( -1 - 2 \beta_{1} + \beta_{4} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{82} + ( 6 + 6 \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{16} + \beta_{17} ) q^{83} + ( -2 - 2 \beta_{1} + 3 \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{16} + \beta_{17} ) q^{85} + ( 3 \beta_{2} + \beta_{8} - 3 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{18} ) q^{86} -\beta_{19} q^{88} + ( -5 \beta_{2} - 5 \beta_{11} + 2 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{89} + ( 2 \beta_{2} + \beta_{3} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{14} - \beta_{19} ) q^{91} + ( 2 - 2 \beta_{4} - \beta_{5} - \beta_{7} ) q^{92} + ( -\beta_{3} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} - \beta_{19} ) q^{94} + ( 2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} + 2 \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{95} + ( 2 \beta_{2} + 3 \beta_{3} + \beta_{8} - \beta_{10} - \beta_{12} - \beta_{13} + 3 \beta_{14} + 3 \beta_{15} ) q^{97} + ( -\beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{18} + 2 \beta_{19} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 4q^{5} - 4q^{7} + O(q^{10})$$ $$20q + 4q^{5} - 4q^{7} + 8q^{11} - 20q^{16} - 4q^{17} - 44q^{23} + 4q^{25} + 8q^{26} - 4q^{28} - 4q^{35} + 4q^{38} + 52q^{43} - 4q^{47} + 16q^{55} + 8q^{58} + 32q^{61} + 8q^{62} - 4q^{68} - 20q^{73} + 20q^{76} + 24q^{77} - 4q^{80} - 24q^{82} + 116q^{83} - 60q^{85} + 44q^{92} + 32q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 153 x^{16} + 6416 x^{12} + 78648 x^{8} + 19120 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-16765 \nu^{16} - 2625903 \nu^{12} - 115242314 \nu^{8} - 1515721480 \nu^{4} - 599186688$$$$)/ 174141704$$ $$\beta_{2}$$ $$=$$ $$($$$$806911 \nu^{17} + 123405389 \nu^{13} + 5168924442 \nu^{9} + 63121710948 \nu^{5} + 12328816792 \nu$$$$)/ 2089700448$$ $$\beta_{3}$$ $$=$$ $$($$$$-604825 \nu^{17} - 92597906 \nu^{13} - 3887923677 \nu^{9} - 47719026672 \nu^{5} - 10577293948 \nu$$$$)/ 522425112$$ $$\beta_{4}$$ $$=$$ $$($$$$54129 \nu^{18} + 8282406 \nu^{14} + 347394797 \nu^{10} + 4261561356 \nu^{6} + 1096207364 \nu^{2}$$$$)/31662128$$ $$\beta_{5}$$ $$=$$ $$($$$$11884471 \nu^{18} - 808344 \nu^{16} + 1818129965 \nu^{14} - 123229932 \nu^{12} + 76223127354 \nu^{10} - 5124003060 \nu^{8} + 933673009644 \nu^{6} - 61610737104 \nu^{4} + 216220477192 \nu^{2} - 7006091376$$$$)/ 4179400896$$ $$\beta_{6}$$ $$=$$ $$($$$$-11884471 \nu^{18} - 808344 \nu^{16} - 1818129965 \nu^{14} - 123229932 \nu^{12} - 76223127354 \nu^{10} - 5124003060 \nu^{8} - 933673009644 \nu^{6} - 61610737104 \nu^{4} - 216220477192 \nu^{2} - 7006091376$$$$)/ 4179400896$$ $$\beta_{7}$$ $$=$$ $$($$$$13498293 \nu^{18} - 604298 \nu^{16} + 2064940743 \nu^{14} - 92070010 \nu^{12} + 86560976238 \nu^{10} - 3830938056 \nu^{8} + 1059916431540 \nu^{6} - 46444027032 \nu^{4} + 240878110776 \nu^{2} - 6281901056$$$$)/ 4179400896$$ $$\beta_{8}$$ $$=$$ $$($$$$-12734103 \nu^{18} + 1699264 \nu^{17} - 201180 \nu^{16} - 1948517589 \nu^{14} + 260775248 \nu^{13} - 31510836 \nu^{12} - 81728941602 \nu^{10} + 11011628496 \nu^{9} - 1382907768 \nu^{8} - 1002348912780 \nu^{6} + 137351806272 \nu^{5} - 18188657760 \nu^{4} - 248442762312 \nu^{2} + 64444570240 \nu - 7190240256$$$$)/ 4179400896$$ $$\beta_{9}$$ $$=$$ $$($$$$-13498293 \nu^{18} - 604298 \nu^{16} - 2064940743 \nu^{14} - 92070010 \nu^{12} - 86560976238 \nu^{10} - 3830938056 \nu^{8} - 1059916431540 \nu^{6} - 46444027032 \nu^{4} - 240878110776 \nu^{2} - 6281901056$$$$)/ 4179400896$$ $$\beta_{10}$$ $$=$$ $$($$$$-12734103 \nu^{18} - 1699264 \nu^{17} - 201180 \nu^{16} - 1948517589 \nu^{14} - 260775248 \nu^{13} - 31510836 \nu^{12} - 81728941602 \nu^{10} - 11011628496 \nu^{9} - 1382907768 \nu^{8} - 1002348912780 \nu^{6} - 137351806272 \nu^{5} - 18188657760 \nu^{4} - 248442762312 \nu^{2} - 64444570240 \nu - 7190240256$$$$)/ 4179400896$$ $$\beta_{11}$$ $$=$$ $$($$$$-45261895 \nu^{19} - 6924865889 \nu^{15} - 290369158398 \nu^{11} - 3558464452956 \nu^{7} - 850240722328 \nu^{3}$$$$)/ 4179400896$$ $$\beta_{12}$$ $$=$$ $$($$$$-21963356 \nu^{19} + 518651 \nu^{17} - 3360431092 \nu^{15} + 79345865 \nu^{13} - 140922857856 \nu^{11} + 3327761602 \nu^{9} - 1727640346328 \nu^{7} + 40901095068 \nu^{5} - 423306773744 \nu^{3} + 13651993928 \nu$$$$)/ 1393133632$$ $$\beta_{13}$$ $$=$$ $$($$$$21963356 \nu^{19} + 518651 \nu^{17} + 3360431092 \nu^{15} + 79345865 \nu^{13} + 140922857856 \nu^{11} + 3327761602 \nu^{9} + 1727640346328 \nu^{7} + 40901095068 \nu^{5} + 423306773744 \nu^{3} + 13651993928 \nu$$$$)/ 1393133632$$ $$\beta_{14}$$ $$=$$ $$($$$$24345032 \nu^{19} - 518651 \nu^{17} + 3724856956 \nu^{15} - 79345865 \nu^{13} + 156208228924 \nu^{11} - 3327761602 \nu^{9} + 1915149045992 \nu^{7} - 40901095068 \nu^{5} + 471539897760 \nu^{3} - 12258860296 \nu$$$$)/ 1393133632$$ $$\beta_{15}$$ $$=$$ $$($$$$-24345032 \nu^{19} - 518651 \nu^{17} - 3724856956 \nu^{15} - 79345865 \nu^{13} - 156208228924 \nu^{11} - 3327761602 \nu^{9} - 1915149045992 \nu^{7} - 40901095068 \nu^{5} - 471539897760 \nu^{3} - 12258860296 \nu$$$$)/ 1393133632$$ $$\beta_{16}$$ $$=$$ $$($$$$-77025497 \nu^{19} - 2419300 \nu^{17} + 1726694 \nu^{16} - 11784791035 \nu^{15} - 370391624 \nu^{13} + 264708058 \nu^{12} - 494177340558 \nu^{11} - 15551694708 \nu^{9} + 11136819804 \nu^{8} - 6057012474372 \nu^{7} - 190876106688 \nu^{5} + 136345642968 \nu^{4} - 1459603333880 \nu^{3} - 42309175792 \nu + 11065189808$$$$)/ 4179400896$$ $$\beta_{17}$$ $$=$$ $$($$$$-77025497 \nu^{19} - 22270858 \nu^{18} - 2419300 \nu^{17} - 11784791035 \nu^{15} - 3406995518 \nu^{14} - 370391624 \nu^{13} - 494177340558 \nu^{11} - 142817533980 \nu^{10} - 15551694708 \nu^{9} - 6057012474372 \nu^{7} - 1748182870776 \nu^{6} - 190876106688 \nu^{5} - 1459603333880 \nu^{3} - 379366025296 \nu^{2} - 42309175792 \nu$$$$)/ 4179400896$$ $$\beta_{18}$$ $$=$$ $$($$$$113542400 \nu^{19} + 12734103 \nu^{18} + 201180 \nu^{16} + 17372793436 \nu^{15} + 1948517589 \nu^{14} + 31510836 \nu^{12} + 728609156748 \nu^{11} + 81728941602 \nu^{10} + 1382907768 \nu^{8} + 8934778735776 \nu^{7} + 1002348912780 \nu^{6} + 18188657760 \nu^{4} + 2227441426544 \nu^{3} + 248442762312 \nu^{2} + 7190240256$$$$)/ 4179400896$$ $$\beta_{19}$$ $$=$$ $$($$$$77025497 \nu^{19} + 11784791035 \nu^{15} + 494177340558 \nu^{11} + 6057012474372 \nu^{7} + 1459603333880 \nu^{3}$$$$)/ 2089700448$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{19} - 2 \beta_{17} + 2 \beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 8 \beta_{4} + \beta_{3} - 2 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{19} - 2 \beta_{18} - 7 \beta_{15} + 7 \beta_{14} - 7 \beta_{13} + 7 \beta_{12} + 4 \beta_{11} - \beta_{10} - \beta_{8}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{19} - 18 \beta_{16} - 3 \beta_{9} - 3 \beta_{7} - 13 \beta_{6} - 13 \beta_{5} + 9 \beta_{3} - 16 \beta_{1} - 60$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-51 \beta_{15} - 51 \beta_{14} - 59 \beta_{13} - 59 \beta_{12} - 13 \beta_{10} + 13 \beta_{8} + 28 \beta_{3} + 72 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$79 \beta_{19} + 158 \beta_{17} - 136 \beta_{10} - 143 \beta_{9} - 136 \beta_{8} + 143 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} - 496 \beta_{4} - 79 \beta_{3} + 136 \beta_{1}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-312 \beta_{19} + 294 \beta_{18} + 531 \beta_{15} - 531 \beta_{14} + 395 \beta_{13} - 395 \beta_{12} - 888 \beta_{11} + 147 \beta_{10} + 147 \beta_{8}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$707 \beta_{19} + 1414 \beta_{16} - 343 \beta_{9} - 343 \beta_{7} + 1463 \beta_{6} + 1463 \beta_{5} - 707 \beta_{3} + 1220 \beta_{1} + 4328$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$3235 \beta_{15} + 3235 \beta_{14} + 4947 \beta_{13} + 4947 \beta_{12} + 1563 \beta_{10} - 1563 \beta_{8} - 3220 \beta_{3} - 9664 \beta_{2}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-6455 \beta_{19} - 12910 \beta_{17} + 11308 \beta_{10} + 14507 \beta_{9} + 11308 \beta_{8} - 14507 \beta_{7} - 4723 \beta_{6} + 4723 \beta_{5} + 39168 \beta_{4} + 6455 \beta_{3} - 11308 \beta_{1}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$32140 \beta_{19} - 32062 \beta_{18} - 46923 \beta_{15} + 46923 \beta_{14} - 27771 \beta_{13} + 27771 \beta_{12} + 99536 \beta_{11} - 16031 \beta_{10} - 16031 \beta_{8}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-59911 \beta_{19} - 119822 \beta_{16} + 54059 \beta_{9} + 54059 \beta_{7} - 141819 \beta_{6} - 141819 \beta_{5} + 59911 \beta_{3} - 106756 \beta_{1} - 363056$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-247291 \beta_{15} - 247291 \beta_{14} - 449099 \beta_{13} - 449099 \beta_{12} - 160815 \beta_{10} + 160815 \beta_{8} + 315700 \beta_{3} + 997024 \beta_{2}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$562991 \beta_{19} + 1125982 \beta_{17} - 1018020 \beta_{10} - 1377203 \beta_{9} - 1018020 \beta_{8} + 1377203 \beta_{7} + 572851 \beta_{6} - 572851 \beta_{5} - 3416960 \beta_{4} - 562991 \beta_{3} + 1018020 \beta_{1}$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-3076036 \beta_{19} + 3181742 \beta_{18} + 4317371 \beta_{15} - 4317371 \beta_{14} + 2261611 \beta_{13} - 2261611 \beta_{12} - 9836256 \beta_{11} + 1590871 \beta_{10} + 1590871 \beta_{8}$$$$)/2$$ $$\nu^{16}$$ $$=$$ $$($$$$5337647 \beta_{19} + 10675294 \beta_{16} - 5838259 \beta_{9} - 5838259 \beta_{7} + 13331811 \beta_{6} + 13331811 \beta_{5} - 5337647 \beta_{3} + 9760724 \beta_{1} + 32468000$$$$)/2$$ $$\nu^{17}$$ $$=$$ $$($$$$21071035 \beta_{15} + 21071035 \beta_{14} + 41593707 \beta_{13} + 41593707 \beta_{12} + 15598983 \beta_{10} - 15598983 \beta_{8} - 29845364 \beta_{3} - 96201728 \beta_{2}$$$$)/2$$ $$\nu^{18}$$ $$=$$ $$($$$$-50916399 \beta_{19} - 101832798 \beta_{17} + 93862708 \beta_{10} + 128862627 \beta_{9} + 93862708 \beta_{8} - 128862627 \beta_{7} - 58227795 \beta_{6} + 58227795 \beta_{5} + 310349696 \beta_{4} + 50916399 \beta_{3} - 93862708 \beta_{1}$$$$)/2$$ $$\nu^{19}$$ $$=$$ $$($$$$288923220 \beta_{19} - 304181006 \beta_{18} - 401128059 \beta_{15} + 401128059 \beta_{14} - 198779851 \beta_{13} + 198779851 \beta_{12} + 936087104 \beta_{11} - 152090503 \beta_{10} - 152090503 \beta_{8}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.498616 − 0.498616i 2.19691 − 2.19691i −1.75036 + 1.75036i −0.120370 + 0.120370i −1.53190 + 1.53190i −0.498616 + 0.498616i −2.19691 + 2.19691i 1.75036 − 1.75036i 0.120370 − 0.120370i 1.53190 − 1.53190i 0.498616 + 0.498616i 2.19691 + 2.19691i −1.75036 − 1.75036i −0.120370 − 0.120370i −1.53190 − 1.53190i −0.498616 − 0.498616i −2.19691 − 2.19691i 1.75036 + 1.75036i 0.120370 + 0.120370i 1.53190 + 1.53190i
−0.707107 + 0.707107i 0 1.00000i −1.89390 + 1.18875i 0 0.705149 + 0.705149i 0.707107 + 0.707107i 0 0.498616 2.17977i
37.2 −0.707107 + 0.707107i 0 1.00000i −1.25884 1.84806i 0 3.10690 + 3.10690i 0.707107 + 0.707107i 0 2.19691 + 0.416642i
37.3 −0.707107 + 0.707107i 0 1.00000i 0.253765 + 2.22162i 0 −2.47539 2.47539i 0.707107 + 0.707107i 0 −1.75036 1.39149i
37.4 −0.707107 + 0.707107i 0 1.00000i 1.66396 1.49373i 0 −0.170229 0.170229i 0.707107 + 0.707107i 0 −0.120370 + 2.23283i
37.5 −0.707107 + 0.707107i 0 1.00000i 2.23502 0.0685835i 0 −2.16643 2.16643i 0.707107 + 0.707107i 0 −1.53190 + 1.62889i
37.6 0.707107 0.707107i 0 1.00000i −1.89390 + 1.18875i 0 0.705149 + 0.705149i −0.707107 0.707107i 0 −0.498616 + 2.17977i
37.7 0.707107 0.707107i 0 1.00000i −1.25884 1.84806i 0 3.10690 + 3.10690i −0.707107 0.707107i 0 −2.19691 0.416642i
37.8 0.707107 0.707107i 0 1.00000i 0.253765 + 2.22162i 0 −2.47539 2.47539i −0.707107 0.707107i 0 1.75036 + 1.39149i
37.9 0.707107 0.707107i 0 1.00000i 1.66396 1.49373i 0 −0.170229 0.170229i −0.707107 0.707107i 0 0.120370 2.23283i
37.10 0.707107 0.707107i 0 1.00000i 2.23502 0.0685835i 0 −2.16643 2.16643i −0.707107 0.707107i 0 1.53190 1.62889i
1063.1 −0.707107 0.707107i 0 1.00000i −1.89390 1.18875i 0 0.705149 0.705149i 0.707107 0.707107i 0 0.498616 + 2.17977i
1063.2 −0.707107 0.707107i 0 1.00000i −1.25884 + 1.84806i 0 3.10690 3.10690i 0.707107 0.707107i 0 2.19691 0.416642i
1063.3 −0.707107 0.707107i 0 1.00000i 0.253765 2.22162i 0 −2.47539 + 2.47539i 0.707107 0.707107i 0 −1.75036 + 1.39149i
1063.4 −0.707107 0.707107i 0 1.00000i 1.66396 + 1.49373i 0 −0.170229 + 0.170229i 0.707107 0.707107i 0 −0.120370 2.23283i
1063.5 −0.707107 0.707107i 0 1.00000i 2.23502 + 0.0685835i 0 −2.16643 + 2.16643i 0.707107 0.707107i 0 −1.53190 1.62889i
1063.6 0.707107 + 0.707107i 0 1.00000i −1.89390 1.18875i 0 0.705149 0.705149i −0.707107 + 0.707107i 0 −0.498616 2.17977i
1063.7 0.707107 + 0.707107i 0 1.00000i −1.25884 + 1.84806i 0 3.10690 3.10690i −0.707107 + 0.707107i 0 −2.19691 + 0.416642i
1063.8 0.707107 + 0.707107i 0 1.00000i 0.253765 2.22162i 0 −2.47539 + 2.47539i −0.707107 + 0.707107i 0 1.75036 1.39149i
1063.9 0.707107 + 0.707107i 0 1.00000i 1.66396 + 1.49373i 0 −0.170229 + 0.170229i −0.707107 + 0.707107i 0 0.120370 + 2.23283i
1063.10 0.707107 + 0.707107i 0 1.00000i 2.23502 + 0.0685835i 0 −2.16643 + 2.16643i −0.707107 + 0.707107i 0 1.53190 + 1.62889i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1063.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.p.d 20
3.b odd 2 1 570.2.m.a 20
5.c odd 4 1 inner 1710.2.p.d 20
15.e even 4 1 570.2.m.a 20
19.b odd 2 1 inner 1710.2.p.d 20
57.d even 2 1 570.2.m.a 20
95.g even 4 1 inner 1710.2.p.d 20
285.j odd 4 1 570.2.m.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.m.a 20 3.b odd 2 1
570.2.m.a 20 15.e even 4 1
570.2.m.a 20 57.d even 2 1
570.2.m.a 20 285.j odd 4 1
1710.2.p.d 20 1.a even 1 1 trivial
1710.2.p.d 20 5.c odd 4 1 inner
1710.2.p.d 20 19.b odd 2 1 inner
1710.2.p.d 20 95.g even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{10} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1710, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{5}$$
$3$ $$T^{20}$$
$5$ $$( 3125 - 1250 T + 125 T^{2} - 200 T^{3} + 110 T^{4} - 52 T^{5} + 22 T^{6} - 8 T^{7} + T^{8} - 2 T^{9} + T^{10} )^{2}$$
$7$ $$( 128 + 640 T + 1600 T^{2} - 1600 T^{3} + 1120 T^{4} + 864 T^{5} + 320 T^{6} + 8 T^{7} + 2 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$11$ $$( 272 + 264 T + 8 T^{2} - 32 T^{3} - 2 T^{4} + T^{5} )^{4}$$
$13$ $$4096 + 16228608 T^{4} + 15397632 T^{8} + 1634400 T^{12} + 2656 T^{16} + T^{20}$$
$17$ $$( 86528 + 153088 T + 135424 T^{2} - 12032 T^{3} + 3776 T^{4} + 4224 T^{5} + 2400 T^{6} - 16 T^{7} + 2 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$19$ $$6131066257801 + 917112404214 T^{2} + 89810586829 T^{4} + 5697634120 T^{6} + 301086274 T^{8} + 14825156 T^{10} + 834034 T^{12} + 43720 T^{14} + 1909 T^{16} + 54 T^{18} + T^{20}$$
$23$ $$( 128 + 3968 T + 61504 T^{2} - 6848 T^{3} + 864 T^{4} + 7648 T^{5} + 5120 T^{6} + 1464 T^{7} + 242 T^{8} + 22 T^{9} + T^{10} )^{2}$$
$29$ $$( -9193472 + 2065984 T^{2} - 175872 T^{4} + 7088 T^{6} - 136 T^{8} + T^{10} )^{2}$$
$31$ $$( 9193472 + 2065984 T^{2} + 175872 T^{4} + 7088 T^{6} + 136 T^{8} + T^{10} )^{2}$$
$37$ $$5035857504178176 + 401284665540864 T^{4} + 436352961280 T^{8} + 154337888 T^{12} + 21472 T^{16} + T^{20}$$
$41$ $$( 700928 + 2571536 T^{2} + 336608 T^{4} + 13864 T^{6} + 216 T^{8} + T^{10} )^{2}$$
$43$ $$( 12500000 - 14500000 T + 8410000 T^{2} - 2510400 T^{3} + 431088 T^{4} - 40048 T^{5} + 8504 T^{6} - 2128 T^{7} + 338 T^{8} - 26 T^{9} + T^{10} )^{2}$$
$47$ $$( 1520768 - 2190464 T + 1577536 T^{2} + 1777600 T^{3} + 755424 T^{4} + 163392 T^{5} + 19936 T^{6} + 968 T^{7} + 2 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$53$ $$7728058388709376 + 160876543672320 T^{4} + 417306124288 T^{8} + 211288576 T^{12} + 33488 T^{16} + T^{20}$$
$59$ $$( -2196608 + 819856 T^{2} - 106080 T^{4} + 6008 T^{6} - 144 T^{8} + T^{10} )^{2}$$
$61$ $$( -320 + 80 T + 112 T^{2} - 16 T^{3} - 8 T^{4} + T^{5} )^{4}$$
$67$ $$17592186044416 + 82119774699520 T^{4} + 1319494483968 T^{8} + 420478976 T^{12} + 39168 T^{16} + T^{20}$$
$71$ $$( 22151168 + 9781248 T^{2} + 702464 T^{4} + 19456 T^{6} + 232 T^{8} + T^{10} )^{2}$$
$73$ $$( 22957088 + 29136800 T + 18490000 T^{2} + 6939392 T^{3} + 1659088 T^{4} + 242448 T^{5} + 20264 T^{6} + 704 T^{7} + 50 T^{8} + 10 T^{9} + T^{10} )^{2}$$
$79$ $$( -8192 + 167488 T^{2} - 134784 T^{4} + 9968 T^{6} - 184 T^{8} + T^{10} )^{2}$$
$83$ $$( 14623232 - 51657216 T + 91240704 T^{2} - 49133312 T^{3} + 14745408 T^{4} - 2777280 T^{5} + 350560 T^{6} - 29776 T^{7} + 1682 T^{8} - 58 T^{9} + T^{10} )^{2}$$
$89$ $$( -62720000 + 25805200 T^{2} - 2898528 T^{4} + 63560 T^{6} - 472 T^{8} + T^{10} )^{2}$$
$97$ $$439242249480503296 + 70710927395491840 T^{4} + 23912789004288 T^{8} + 2551174016 T^{12} + 94128 T^{16} + T^{20}$$